William Chen

Jozsef Beck

Yuxuan Yang

The subject of this monograph is to describe orbits of slowly chaotic motion. The study of geodesic flow on the unit torus is motivated by the irrational rotation sequence, where the most outstanding result is the Kronecker-Weyl equidistribution theorem and its time-quantitative enhancements, including superuniformity. Another important result is the Khinchin density theorem on superdensity, a best possible form of time-quantitative density. The purpose of this monograph is to extend these classical time-quantitative results to some non-integrable flat dynamical systems.The theory of dynamical systems is on the most part about the qualitative behavior of typical orbits and not about individual orbits. Thus, our study deviates from, and indeed is in complete contrast to, what is considered the mainstream research in dynamical systems. We establish non-trivial results concerning explicit individual orbits and describe their long-term behavior in a precise time-quantitative way. Our non-ergodic approach gives rise to a few new methods. These are based on a combination of ideas in combinatorics, number theory, geometry and linear algebra.Approximately half of this monograph is devoted to a time-quantitative study of two concrete simple non-integrable flat dynamical systems. The first concerns billiard in the L-shape region which is equivalent to geodesic flow on the L-surface. The second concerns geodesic flow on the surface of the unit cube. In each, we give a complete description of time-quantitative equidistribution for every geodesic with a quadratic irrational slope.

Non-integrable Dynamics: Time-quantitative Results

We know very little about the time-evolution of many-particle dynamical systems, the subject of our book. Even the 3-body problem has no explicit solution (we cannot solve the corresponding system of differential equations, and computer simulation indicates hopelessly chaotic behaviour). For example, what can we say about the typical time evolution of a large system starting from a stage far from equilibrium? What happens in a realistic time scale? The reader's first reaction is probably: What about the famous Second Law (of thermodynamics)?Unfortunately, there are plenty of notorious mathematical problems surrounding the Second Law. (1) How to rigorously define entropy? How to convert the well known intuitions (like 'disorder' and 'energy spreading') into precise mathematical definitions? (2) How to express the Second Law in forms of a rigorous mathematical theorem? (3) The Second Law is a 'soft' qualitative statement about entropy increase, but does not say anything about the necessary time to reach equilibrium.The object of this book is to answer questions (1)-(2)-(3). We rigorously prove a Time-Quantitative Second Law that works on a realistic time scale. As a by product, we clarify the Loschmidt-paradox and the related reversibility/irreversibility paradox.

Equidistribution Of Dynamical Systems: Time-quantitative Second Law

József Beck

Traditional game theory has been successful at developing strategy in games of incomplete information: when one player knows something that the other does not. But it has little to say about games of complete information, for example tic-tac-toe, solitaire and hex. This is the subject of combinatorial game theory. Most board games are a challenge for mathematics: to analyze a position one has to examine the available options, and then the further options available after selecting any option, and so on. This leads to combinatorial chaos, where brute force study is impractical. In this comprehensive volume, József Beck shows readers how to escape from the combinatorial chaos via the fake probabilistic method, a game-theoretic adaptation of the probabilistic method in combinatorics. Using this, the author is able to determine exact results about infinite classes of many games, leading to the discovery of some striking new duality principles.

Jozsef Beck

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